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Fungrim entry: 103bfb

θ3 ⁣(z+τ,τ)=eπi(2z+τ)θ3 ⁣(z,τ)\theta_{3}\!\left(z + \tau , \tau\right) = {e}^{-\pi i \left(2 z + \tau\right)} \theta_{3}\!\left(z , \tau\right)
Assumptions:zCandτHz \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \tau \in \mathbb{H}
\theta_{3}\!\left(z + \tau , \tau\right) = {e}^{-\pi i \left(2 z + \tau\right)} \theta_{3}\!\left(z , \tau\right)

z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \tau \in \mathbb{H}
Fungrim symbol Notation Short description
JacobiThetaθj ⁣(z,τ)\theta_{j}\!\left(z , \tau\right) Jacobi theta function
Expez{e}^{z} Exponential function
Piπ\pi The constant pi (3.14...)
ConstIii Imaginary unit
CCC\mathbb{C} Complex numbers
HHH\mathbb{H} Upper complex half-plane
Source code for this entry:
    Formula(Equal(JacobiTheta(3, Add(z, tau), tau), Mul(Exp(Neg(Mul(Mul(Pi, ConstI), Add(Mul(2, z), tau)))), JacobiTheta(3, z, tau)))),
    Variables(z, tau),
    Assumptions(And(Element(z, CC), Element(tau, HH))))

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Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2020-01-31 18:09:28.494564 UTC